Gradual Parametricity, Revisited

Transcription

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sha1_base64="dv+za/citurtsmyynlkk2y0xpa=">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</latexit> <latexit sha1_base64="rcjy0lef1cmeipbmve2q4kqbleg=">aaacjhicbvc7tsmwfhv4lvikmhaxqjbyqbiwmkccbbyi0yfuvjjok1vx45sp1ivdebnyi/yemmlhwe4ctbglajmt5+nx7dk+phzoqton8wevlk6tr66wn8ubw9s6uvbffuikrmdsxyej2fkqio5w0ndwmdgjjuoqz0vzh11m9pszsuchv9sqmvqgnoa0prtpifbvi3inty6gdwk9rapszl5qrc+vqtg64snycvegbrt/+8gkbk4hwjrlsqus6se6lsgqkgzmwvusrgoergpcuorxfrps/bntegsuaizcmmnwytfjhrfsk0i33rgsa/vfc0t/6t1ex2e91lk40qtjmedworblwcwcayojfiziseis2p2hiijmla5fb2cmmakjbymmwmpj/aokdtsgnjny9kkbroa65tc++car1exfucfaijoelzkad3iagaaimhsateayv1qp1ar1z77pwjavwhia/sd6/abqmo04=</latexit> Gradual Typing 101 precision Int! Int v Int!? v?!? v? Gradual Parametricity, Revisited some gradual types convey more information than consistency others best-effort static checking backed by dynamic checking Int!??! Bool Int!? 6 Bool!? <latexit sha1_base64="xwkkkm4momkpbjatwvvkw8bug=">aaackhicbzdnsgmxfiv+fvr9wvudbybdewgte60qib3anywuiukslkndstdemmuiyuxfrrb6ho3hri/gezqykwj0qojx7l/fmcxlothhdn2dicmp6zry0v55fwfxarqysnrvmfaenirlurqbrypmgdcmmp61eurwhnf4hvzo8ft2nsjmprswgoz0y3wgwmyknjbqvtf9mglr7ipwj5atpfm1i5b9lyudht1j1a24h9nd46za7+/rapaebfy4yespdevhncsddtze9pjsdkmcdos+6mmcsy9fepb1gocu93jio8m0bznqhrjzz89q0h/tmq41noqb7yzxuzwj9fy8l9aozrqsdjikknfws0keo5mhllvfdifcwgd6zbrdf7kyk3wgfilluywxmiy2yydk3pfzo+iebli0kbxzj9pcq3luzbuwthjuuuqwavuwax7sqx1o4rwaqoaohuejnp0h58v5dd5grrpo18wa/jlz/glynkav</latexit> sha1_base64="gi5b4mjtwakzfmc2wyu/lu7/bly=">aaackhicbvdlsgmxfm3uv62vqitxybaibiwzbnslrte6u7fw6jsaydyxozlksdkfmnqp/kvd6n+4e7d+gl9gplwdsbwooce7s0jes60cd1pzaxotu9u5wtzc0vlc6vl1eutewvhtqvkrrggjgteddmmpholfa4obdi+ge536jb0ozks5np4fwtg4fixglxkrt8oz/kgzeocd+ifafnl5mmfapporjsv2uufv3bpyej+kulu7v7i8gcjzu/zuh5kmmqhdodg66bmjawveguy5dep+qiehtetuowmpidhovjb6yabvwsekvt22bng6vdermkt+3fgj2niovq3l4v/ec3urputjikknsdoefgucmwkzlvbivnade9bqqhi9lzmo0qramx3j8uzekimgb5b3rwpfvcnijzkrzflfwlv7tvz61657atghqjinbrjtpghtpdnsczladushhtajenkgzrpz4ryorwvoz2yv/ydz9ggop6gy</latexit> sha1_base64="dmdpmor1sfp38tc/jeovjrbdvfu=">aaackhicbvdlsgmxfm34rpu16tkfwsk4scy40zuw3eiugn1ap5rmjtog5jekmuizuvrndkv/4u669rp8atn2bg09ediccw/35oqjo9p43thzwl5zvsvbbq3t7z3dt29/bqwqckkhiwtqhkitrgvpgaoyaszkij4yegj7n9o/maake2ledtdhlq56goau4ymltruuavddy7gse1diq0gaycbjdsspnycute2zsclhi/jywqo9pxv4ji4pqtytbdwrd8lzhtdcldmsojypbqkidcr13sslqgtnq7m35kbe+sesfykvvswvp1dyjdoshd+0kr6an572j+j/sk182c6osfjdbj4tilmgjystvmbefcggds1bwff7k8q9pba2trtima1meympojpe9ohhg0r0vlql+fovljl6edn3yv6dv6pu8rok4bacg1pggwtqaegcmoagyfwal7bm/psvdsfzng2uutkmqpwb87nnzzjpsm=</latexit> Éric Tanter Inria Paris & University of Chile joint work with Matías Toro & Elizabeth Labrada type safety (+ s) conservative extension of static language embedding of dynamic language gradual guarantees SGG & DGG 1 2 Challenging Gradual Typing Parametric Polymorphism subtyping unions recursive types refinements effects session types 3 typestates parametric polymorphism ownership security typing System F explicit f :.! f = Λ.λx:.x f [Int] 10 f [Bool] add type binders infer type instantiations Haskell implicit f :! f = λx.x f 10 f implicit polymorphism induces a notion of subtyping 8.! <: Int! Int add [Int] Int! Int <: 8.Int! Int 4 add Λ

2 Parametricity 101 Gradual Parametricity Flavors of parametric polymorphism g : (.!)! Int g = λh:(.!).h [Int] 10 Genericity: type safety with generic definitions Parametricity: strong reasoning principle ( theorems for free ) G f : 8.! P n0 <latexit sha1_base64="6pcm44zr9pqg/kviojpilr4d0e=">aaab+3icbvdlssnafl2pr1pfvzdugkvwvrirdfl047if+4a2lmnkph06myszivbcv8ctfoe7cevh+ah+h5m0c209mha45x7unemnncntof9wzwnza3unulvb2z84pkofn/runeqkrrzwa58opazgv3nnmdbipfepse+p7vp/f4tssvi8ajncormqgwmkq0ktpig84taeavu7ckjsgrhtc/x4fmu0jfjpyottqdrltzurqrjkuaqnuyulojexwakggesovkw5d2bdgcewwluyjbrfq70rgiqmkw8mi6knatlxf+8yardwy9jikk1crpcfkbc1rgd/9oomesq+dwqqiuzt9p0sish2nrtgxblmcqczb3ohamf3e1hsu2q6ttptddad2vdvtidc7gef26gbq/qhi5qqhigf3i1ftab9w59lecrvpk5ht+wpn8agoavha==</latexit> <latexit Gradual parametricity is challenging early work on PBC [POPL 11] without proof of parametricity several recent developments: [ICFP 17], System FG [ICFP 17], ie et al [ESOP 18] arguable decisions, problems, conjectures (dynamic GG?) free theorems not fully understood n <latexit sha1_base64="6pcm44zr9pqg/kviojpilr4d0e=">aaab+3icbvdlssnafl2pr1pfvzdugkvwvrirdfl047if+4a2lmnkph06myszivbcv8ctfoe7cevh+ah+h5m0c209mha45x7unemnncntof9wzwnza3unulvb2z84pkofn/runeqkrrzwa58opazgv3nnmdbipfepse+p7vp/f4tssvi8ajncormqgwmkq0ktpig84taeavu7ckjsgrhtc/x4fmu0jfjpyottqdrltzurqrjkuaqnuyulojexwakggesovkw5d2bdgcewwluyjbrfq70rgiqmkw8mi6knatlxf+8yardwy9jikk1crpcfkbc1rgd/9oomesq+dwqqiuzt9p0sish2nrtgxblmcqczb3ohamf3e1hsu2q6ttptddad2vdvtidc7gef26gbq/qhi5qqhigf3i1ftab9w59lecrvpk5ht+wpn8agoavha==</latexit> <latexit <latexit sha1_base64="/ytveqe+g2lilrwwqwvvan32i8=">aaab/hicbvdlssnafl2pr1pfvzdugkv0vrirdfl047kkfuabymr60w6dtmlmrcihfofb/qj34tz/8qp8dydpftp6yobwzj3co8epovpacb6s0srq2vpgeboytb2zu1fdp2irkjeuwztikez6rcfnaluaay7dwcijfy4df3kt+z1hlipf4kfpy/rcmhisyjroi92l00g15tsdhpyycqtsgwlnqfw7p4xoeqlqlboleq4tay8lujpkcvbpjwpjqidkhd1dbqlrewl+6cw+mcrqdijpntb2rv5opcruahr6zjikeqwwvuz8z+slorjyuibirkog80vbwm0d2dm37sgtsdwfgkkozozwm46jjfsbcir9pjgomwcczcwomenfwxhmbtp665td+8uao3roqeyhmexniell9caw2hccyge8awv8go9ww/wu/uxhy1zreyq/sd6/ahktjw1</latexit> <latexit sha1_base64="vuomklr0yzmectbbhlvyyfzpds=">aaacbhicbvdlssnafl3xweur6tlnybfcluqebtdfn7qryb+yhdkzttqhk0mymqgldosunuvccdu/q8/wp9wkmahrqcgdufcw71zgoqzpw37y1palldw69svde3tnd2a3v7hrwnkta2iksewfwldnb25pptnujpdgkoo0g4+vc7z5sqvgs7vuko6eh4kfjgbtpifqu0sudyu036/v7yzdac0spyr1knhq1769quzsiapnofbkdexe+xmwmhfop1uvvttbziyh1dvu4igqpysunqjjowxqgevzheaf+jur4uipsrsyyqjrkzr3cve/z011eofntcsppolmfouprzpg+ffrgelknj8ygolk5lzerlhiok1jva8izgmamshygttu9olmt7bioqcnx244d2f15lzuauo4qhowifzamintkanbaq8wwu8wk/wm/vufcxgl6wycwb/yh3+aitommw=</latexit> g : 8Y.! Y! <latexit sha1_base64="jyzed7aeytaet1dnl6wzunbalgs=">aaacfnicbvbps8mwhe39o+e/qje9bifgxdgkohiqorepe9zwszarpukwlqylsyvrbn4mp4f/qtexktp4dfw7trqtcfjdze+z3yy/mtrqwyrc9jyfpeww1slzd39jc2jz3dtsytgumlryzwdg+korrtlqkkkacrbau+yx0/nfn7nceija05vdqnbavqgnoq4qr0llf3b9cqjembwimot06dodjfezml9m3a1bdkgdniv2sgijr7jvfbhdjncjcyyak7nlworwmcuuxi5oqm0qsidxca9ltlkoisc8r/jcbr1ojon5eh65gof5ozcischz5ejjcaihnvvz8z+ulkrzwmsqtvbgopw+fkymqhnkhmkccymgmiasqn4v4iescctdw9utgllaqspppmc6f7s2rbmsfu0blt1++6s1rgug6qaa3aijoenzked3iimaaemhsezeagvxppxzrwbh9prbapm7ie/md5/ajusnvw=</latexit> g [Int] [Int] n1 n2 <latexit sha1_base64="/ytveqe+g2lilrwwqwvvan32i8=">aaab/hicbvdlssnafl2pr1pfvzdugkv0vrirdfl047kkfuabymr60w6dtmlmrcihfofb/qj34tz/8qp8dydpftp6yobwzj3co8epovpacb6s0srq2vpgeboytb2zu1fdp2irkjeuwztikez6rcfnaluaay7dwcijfy4df3kt+z1hlipf4kfpy/rcmhisyjroi92l00g15tsdhpyycqtsgwlnqfw7p4xoeqlqlboleq4tay8lujpkcvbpjwpjqidkhd1dbqlrewl+6cw+mcrqdijpntb2rv5opcruahr6zjikeqwwvuz8z+slorjyuibirkog80vbwm0d2dm37sgtsdwfgkkozozwm46jjfsbcir9pjgomwcczcwomenfwxhmbtp665td+8uao3roqeyhmexniell9caw2hccyge8awv8go9ww/wu/uxhy1zreyq/sd6/ahktjw1</latexit> <latexit sha1_base64="yl7a9ak8rb16jmtqxaiywwgrg9w=">aaab/icbvdlssnafl2pr1pfvzdugkvwvrirdfl047kiryu2lmnkph06myszivbc8qvc6he4e7d+ix/gfzhjs9dwawohc+7h3jl+wpnsjvnlvvzw19y3qpu1re2d3b36/kfxamk2kexj2pjwo5e9jrthpsjrjj5hn88cfuf/wifkxwnzrayjereachywsbaq7msh9ybtdaryy8qtsqnktif170eq0zrcosknsvvdj9ferqrmloosnkgvjoroyaj7hgosofky4tszfwkuwa5jaz7qdqh+tmqkumoa+wyyinqsfr1c/m/rpzq89dimklsjopnfycpthdv5v+2asasatw0hvdjzq03hrbkqttu1qrhmagyzyhkzamz6crdbwcbds6brnn3b80brqmyockdwdkfgwgw04aba0aeki3igf3i1nqw36936mi9wrdjzch9gff4asrcwka==</latexit> <latexit sha1_base64="9jsurbfu5vldre8bfjj2e084cm0=">aaab/icbvdlssnafl2pr1pfvzdugkvwvzii6lloxmvf+4a2lmn0ph06myszivbc8qvc6he4e7d+ix/gfzhjs9dwawohc+7h3jl+zjnsjvnlldbwnza3ytuvnd29/ypq4vfhrymk2kyrj2tpjwo5e9jwthpsxrjj6hps+tobzo8+olqseg96fqmkrfgaaneg+ledbvdas2poznsveiwpayfwspq92au0sreosknsvvdj9zesqrmloo8mkguxoroyrj7hgosovls/ns5fwaukr1e0jyh7vz9nuhjqnqs9m1kspreluz+j/t3rw5avmxilgqrelgotborkzf9sjjpfqpjoeumnmrtadeemonu1ubnkwhwhabmuaupti7vcwirpnoquu3fvlmrn66khmpzakzydc5fqhftoqrsojoezudverlerhfryzfasormmfyb9fkdsqewkq==</latexit> <latexit sha1_base64="nsa4ttvvz6i/px6rjayrjslfejs=">aaacdnicbvdlssnafj34rpuv69lnybfcluqehw6kbnrwt6gcwuyuwmhtizhzikw0h/wc9zqf7gtt/6ch+b/og2z0nydwxzouyd7ouhkmdko82wtrk6tb2ywtsrbo7t7+/zbpa2stfjo0yqnshsqbzwjagmmortcsqoohsc0fu7zyavcwr93qcgh+tgwaro0qbqw9bl4d97xbo2vpv/7dtwpotpgzeiwpioknpv2txcmnitbamqjuj3sbwfe6kz5tape5mclnargudpuefiuh4+u32ct4ws4iir5gmnz+rvre5ipczxyczjoodq0zuk/3m9teefs5emmkqdl4oyjjwcz4wgummgwo+norqycytma6jjfsbusreljihedhbplwpienfwxhmbtpaq5tc+/oq42roqesoklh6bs56ai10a1qohai6be9oxf0aj1zb9a79tefbgkzch6a+vzb9arncq=</latexit> n 0 n1 design space is subtle and complex <latexit sha1_base64="yl7a9ak8rb16jmtqxaiywwgrg9w=">aaab/icbvdlssnafl2pr1pfvzdugkvwvrirdfl047kiryu2lmnkph06myszivbc8qvc6he4e7d+ix/gfzhjs9dwawohc+7h3jl+wpnsjvnlvvzw19y3qpu1re2d3b36/kfxamk2kexj2pjwo5e9jrthpsjrjj5hn88cfuf/wifkxwnzrayjereachywsbaq7msh9ybtdaryy8qtsqnktif170eq0zrcosknsvvdj9ferqrmloosnkgvjoroyaj7hgosofky4tszfwkuwa5jaz7qdqh+tmqkumoa+wyyinqsfr1c/m/rpzq89dimklsjopnfycpthdv5v+2asasatw0hvdjzq03hrbkqttu1qrhmagyzyhkzamz6crdbwcbds6brnn3b80brqmyockdwdkfgwgw04aba0aeki3igf3i1nqw36936mi9wrdjzch9gff4asrcwka==</latexit> 5 6 Gradual Parametricity Gradual Implicit Polymorphism [ie et al. ESOP 18] and FG use ad hoc static relations <latexit sha1_base64="hvs+aue5spprb5br4cvahwivzq8=">aaacdicbvdnssnagnzuv1r/uj16wsycf0migp6k6mvjbdsgmla2m027dlmjuxulhd6dt+bvn8cbepuzfadfw02ag7yolawz37ddbcmjutn2l1fbwv1b36hvnra2d3b3zoz+tyazwksle5yin0csmmpjv1hfijskgukak4wusn8/gmrkib8k1t4sdoxglemvjagppnl0oeygy6fnth6rv0h2bltuwscjk4fwmbcp2h+e2fcc5iwhvmsmqby6fkz5fqfdmya3izjcncezqia005ion087l6db5rjys6g35cwvl9nchrlou0dvrljnrylnqf+j83yfr06eeup5kihm8/ijigvqklhwbibcgkttvbwfddfeixeggrvvbdk4n5sclkabgunoldnmuvlknvzhjsy7k7b7wvq4q4bacgrpggavqbrega7oag0fwdf7aq/fkvbnvxsf8tgzumqpwb8bnd7mwmuq=</latexit> explicit polymorphism with (different) flavors of implicitness Int! Int?!? Int! Bool 8.! <latexit sha1_base64="str8cy+tqnqf4bcfmebv1xgl+u=">aaacbnicbvdlssnafj34rpvvdelmsahulikiupkig91vsa9oqplmbtqhk0mymqgldo8unuvccdu/q0/wp9wkmahrregozxzd/dw/iqzpw37y1palldw69svde3tnd2a3v7hrwnkkkbxjywpz8o4exawzpnozdiijhpoeupb3k9+whssvg86ekcksggowmem0o170tgp9e4fwf1op2wy4glwknbhvutmtq+3admkyrce05uarv2in2mii1oxymvtdvkba6jkpogyhibmrlisxtfgyyaiexnm9kknjfjoxesk0i32xgri/uvjat/2n9vieszekmoqdhyotdnwmc4lwagtqdwfgecozcyrpimicdwmpqpbglmaqizypgaml6c+rywqees4dgn5/683rwug6qgq3setpcdllat3aiwaiokevsmtcr9ws9we/wx2x1yso9b+jpwj8/ajuzpq==</latexit> <latexit sha1_base64="i0wgl9otvtt+t/v+sl8ckimqpw=">aaacanicbvdlssnafl2pr1pfvzdubovgxpkioctbdooygn1ae8pkmmmhtizhziku0j1f4fa/wj249uf8ap/dazqfth4yojxzd/fo8rpollbtl6u0srq2vlhergxt7+zuvfcpoipojaftevny9nyskgectjtnpysshkc9r1x7czv/tipwkxencthhorhgowmok1kpua51di7cxqnbsup0dlroniduo0bpuv90gjmlehsyck9v37er7gzaaeu6nftdvnmfkjie0b6jaevvelt87rsdgcvays/oerrn6o5hhsklj5jvjcourwvrm4n9ep9hlzcxkasacjjffkyc6rjnpo8cjinrfgiijpkzwxezyymjnhvv3dyybtrkgs3quvpti7pywjlpnncdu+7c9san0vdzticyzgfby6hcfqgjyq4paml/bqpvlv1rv1mr8twumep7a+vwb2hupq==</latexit> <latexit sha1_base64="evnh0f2rsf65sccgr7zztuhoba=">aaacb3icbvdnsgmxgpzw31r/qh69bivgxbirgp6k1ivektgf6c4lm2bb0gyyjlmhlh0an8crpoe38epj+ac+h9m2b20dcawz35bhwoqzbvz3y1lalldwy9sfde3tnd2s3v7ts1trwidsc5vo8sacizowzddattrfmchp61wej37rueqnjpi3owsgss4l1jecdzwcvxbyddpffjruvjuqexw3anqivfmpawz1lulb78nsrptyqjhwnc8nzfbhpvhhnnx0u81ttaz4j7twcpwthwqtuqp0bfveiisyj5byql+tmq41nouh/yyxmag571c/m/rpca6ddimktrqqayfrslhrqj8adrjihldr5zgopjtisgak0ym3anot4jzj0zmshwjpba7epmrljlmwcvzk97deblamy1ugem4ghpw4akqcan1aacbb3igf3h1npw35935mj4uobpmafyb8/kdk4+zqa==</latexit> <latexit sha1_base64="6uqcw1azmfmc7kwwadhkcnfhla8=">aaacehicbvdnssnagnz4w+tf/ll5wsycf0migp6k6evvfwwbaelzbdbt0s0m7g6egvospoffqjv4tu38af8dzdpdto6sddmfmmoe6smsmb8bc4tlyymptrb6+sbm1be7sdmsscuzaoggjcamkcaoctbvvjlipicgogokgo+vc7z4qiwnc79u4j6mbpxgfcolpb6570wjqixb14iuplme3i1fbnhw3yjoe+cijrahvbf/pbcbgcx4qozjgpsvpl50goihmz1l1mkhtherqqnqycxut6edl+ao+0ekjdqz+uykn+tuqolnicb/oyrmooz71c/m/rzsq68hpk00wrjqcfrrmdkohffdckgmdfxpoglkjucveqcysvhqzulce8jbhltbhltvquzuwk86rzajm25dydnzp1ui1caaowtfwwdloghvqam2awsn4bi/g1gy3ox342n6umbumt3wb8bndxtbnde=</latexit> f = Λ.λx:?.x f = Λ.λx:?.x+1 <latexit sha1_base64="9fdl0lpmtioj2eipf3dagfskiy=">aaacehicbvdnssnagnz4w+tf/ll5wsycf0migujbil48vrbtoalls9m0szebslsrauhl+are9qm8ivffwafwpdykowjrwmiw8w07tjaykpvtfxkli0vlk6u1tfr6xubwtrmz25fjjjbp44qlwg2qjixy0lzumekmgqa4ykqbjg4kv/tahkqjv1fjlpgxgnaauyyulvrmfnqjvsgridhowtcfj1fq7zsn27jlwhnivkqbkrt65rcjjilcveyisl7jp0qp0dcuczipo5lkqqij9ca9dtlkcbsz8v2e3iklrdqdvpxbuv1dyjhsztjoncmvjdoesv4n9el1prhz9tnmakcdz9kmoyvakspoahfqqrntyeyuf1v4ihsccs9gb1rwzmiykop8vycqj3cwzmcedu8uxlefurng8rhaqgqnwci6ba85be9ycfmgddb7bm3gbr8at8wa8gx/t0wwjyuybpza+fwbofjvc</latexit> f [Int] n f = Λ.λx:.x f :? g f 8.! Int <latexit sha1_base64="njlw7iszzxblwekvo9mpwdak0xa=">aaacdnicbvdnssnagnz4w+tfrecvi0wykhe0jmwvisyntae8pms2mbjzhdyow0hfwcbzqe3gtr76cd+b7uelz0nabhwhmg3ayigvuktv+mpawv1b1msb9c2t7z1dc6/rlukmmonghccdzakjhlsuvqx4qacodhgpbembwq/90cepam/v5ou+deachprjjswbmbdixkbgioubv14cgm9q4hztc27bfwktkwaoej7yh57yykzmhcfgzky79ip8nmkfmwmtotejkmk8bgnsv9tjmii/bzspovhwgmhlqefv7bufydyfes5iqn9gsm1kvneif7n9tmvfg55wmmcmezj6kmqzayggyukgwyhnnebzud4v4hatcss9v98pghpkiclpmjad6f2d+huspbuc23luzpqt62qhgjgah+ayooactmataimowoarpim8go8gw/gu/exo10yqsw++apj8wdoe5sx</latexit> 8.!? 8.?! Int <latexit sha1_base64="kqkd7b+lv7phevotpge2v2qjeik=">aaaceicbvdnssnagnz4w+tfvdx5wsycf0sigp606evvfewpnkfstl/apztn2n0ijfqpfakv+gtexktp4ap4hm7thlr1yggy+yydjkg4u9pxvqyfxalldswnl9y3nr297zbao4lrqanoaxbadeawccgpppdu1eaokcdq1gedpxw48gfyvfgx4l4eekl1jiknfg6tr7hhlwjluv7f3hu8uscndneuofunb54nbkeqqec9a397vzimeqhnovgq4zqj9jminamcxmuvvzaqoir96bgqsatkz/l6y3xklb42pcwtgufq70rgiqvguwaui6ihatabip95nvshf37grjjqeht6uzhyrgm82ql3masq+cgqqiuztedeemonouvvtyy9sbkgk3wumozizu7wjxpnlzdp+ren1vq18vcjsadtexcte5qqfbvecnrfggntelerwerdfr3fqyni5yrwyp/yh1+qoxgjx+</latexit> 7 System FG same observations wrt statics of and FG separate consistency from subtyping clean definition of consistent subtyping focus on the statics, uses for runtime 8

3 Gradual Runtime as the target of choice for runtime semantics violates scoping & type instantiation Gradual Runtime as the target of choice for runtime semantics violates scoping & type instantiation let f :? = Λ.λx:.x System F G f [Int] 1 1* 1* (f [Int] 1) + 1 pending cast with out-of-scope variable (1 : Int => α : α =>?) (1 : =>?) 9 10 Gradual Runtime as the target of choice for runtime semantics Instantiations matter! gradual types soundly augment expressiveness violates scoping & type instantiation let f :? = Λ.λx:.x System F G t = λx.x Int! Int STLC Bool! Bool t = λx.(x [Int]) F (.!)! Int!Int ( Y.!Y!)! Y.Int!Y!Int f [Bool] F.! Fω P, (.P )! P Int f [Int] GTLC?!? GF (.?)!? instantiations on? are lost! lack of precision backed by runtime enforcement (t 3) id:.! (t id) 1 1 (t 3) 1 (t id) 11 12

4 Gradual Free Theorems Open Challenges [ICFP 17] includes some free theorems about fully-static signatures What about imprecise type signatures? claims only.? constant or fail [POPL 11], or unusable result [STOP 09,ICFP 17] Proper semantics for gradual explicit polymorphism? (/ FC) adequate as an internal cast language? Parametricity &/vs dynamic gradual guarantee?.? always fail [ICFP 17] id? Gradual free theorems? This Work Proper semantics for gradual explicit polymorphism? GSF: derived from F using AGT (++) (/ FC) adequate as an internal cast language? no: type instantiations matter (GSFε) Parametricity &/vs dynamic gradual guarantee? GSF: Gradual System F Gradual free theorems? incompatible! (weaker property holds) disprove claims, prove cheap theorems 15 16

5 Gradual Parametricity, Revisited Concretization for GSF (Int) = { Int } Exploit methodology to gradualize languages: AGT (G1! G2 ) = { T1! T2 T1 2 (G1 ), T2 2 (G2 ) } effective: refinements, effects, unions, security typing, etc. (?) = { T T 2 Type } <latexit sha1_base64="ojtwjze3agwfemgy0riuk8knuae=">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</latexit> Natural definition of gradual types (syntactic) meaning of gradual types () = { } (8.G) induces precision = { 8.T T 2 (G) } <latexit sha1_base64="scr8tfdnxckfzkkg2u2lksipude=">aaacihicbzdlsgmxgiuzmu9vv24cbmsst2ugrf0ixtcukxgl9apjzp5pw0mmshjfmowl6nbfq934lifwycwvqjaeibwoof/sfifcwfauo6hs7s8srq2tgobm5t7+yw9vabok4vhqaneazaadhamysgyyzdo1farmchftzcjpvwejrmsbw3ows6gvqlixglxka90ihgfp8iqsrtm3x6j0njmvnufcqu1v3irxovjkpo5nqvdkh8y0fsan5utrjucmppsrzrjlkbf9venc6appq8dasqtobjb5qi5pbblikfb2simn6e+njaitrykwk4kygz7vxuf/sc10vu3yzjjdug6vshkotyxhtpaivnadr9zq6hi9q2ydogi1fhmr+ymiuqmcngvht+kw1dlhctjg8eyajpnlc9t+rdzrrtrmuajpcx6icphsjaugw1veduzsjj/smpxh59v5c96no0voboca/zhz+q2zoqjn</latexit> and consistency <latexit sha1_base64="dbauega4/3l8qep3sf2vvitstho=">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</latexit> G1 <latexit 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sha1_base64="ywkzxdkmycmqy0cjlm0icjfo=">aaaccnicbvdlssnafj3uv42vapdugkvwisvxo8ucc11wta9os5lmbtqhk0mymrrkyma1bvu/3ilbv0hwd1y7doukractbwyo59zdvpcifgpbpvdkcwtr6yufdfnjc2t7z3s7l5threg0cahc0bxriy5dbqvdforwjw4djouapzzg+nquga8hs1iaa4agnpivyaen6ou/0sxw7auewfokzi5va+e3r9im9rvdln10vjheabggpew4dqr6crakegap2y0lrjim8aa6mnicgowl+ampdagvz/jdor9vq7+tiq4khisuhoywgoo571m/m/rxmo/6ywur7ectqal/jhzkrsyf1sefuaum2iciad6vosmscbe6bmbh5mppapp1kzmv3rxh5ntv2sm1/jimmevb276lzptmpoiilarwfocdnofnqjaqjbijogo7ra3o07own49l4my4wjfmmjp7aep0g/dkfwg==</latexit> Dynamic semantics, <latexit 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sha1_base64="eab4i925x6v1bev5af9atig3c=">aaacenicbvdlsgmxfl1t7w+qi7dbitqn2gjs4llntzgn1aw0omk2ldk8yqzaplml/qrf6ho3hrd7hz6dovmh0i2nogcdjnhu7n8wpothhddye3tr6xuzfluzs7u0ffa+pmjpkfkenevfit2skwesngwznlzjrbhwow35o+up3xptpvkk78wkpj2bb5kfjgbjpu53gia5zu+d94vltykownajd6clkqo/vkbalv+8asbrcqrvbrcsdydz41nl8kmmjpvugmmsayjpcadiyvwfdds2cnz+jmkgeki2wfngim/k6kwgg9eb6dfngm9bi3ff/zookjr3opk3fiqctzrwhckynq9p8oyioswyewykkyvrwrivaygntsotslpgenmwtthnt2o40dlhvssd5yjaukevhx3ipt21vyy48nmaplmgds6jcldsgaqqieibhehlunwfnxmdj+acreyy/sb5+wywq6cd</latexit> sha1_base64="qjf2z2fsx7dtivsnyne+osllzcg=">aaacenicbvdlsgmxfm3ur+v4qrp0eyxc3zqzqrzckhlfuwd2qfkmpk2ni8hyrtk0l/qrf6ho3hrd/gd4tovmh0i2nogcdjnhu7ncrngtfg8dye3tr6xms9suds7u3v7xypdppapwqsbjzoqhsjngbwkyahhpj0ognjiscsck/91ogotaw4m+oebbz1by0prszknw4fcy7knz3/rfcservvbrhk/aupvwh986oqv6j1il/dsokue2ewq1p3fc8xqyauoziridtnnukqhqi+6vgqecc6ygynt+cpvsiys2wfmhcm/k5kigs95qgd5mgm9li3ff/zoqmjr4kmiiq1rod5ojhl0eg4/t+mqclyslelcctqb4v4gbtcxrbkdmfblcixftakj78akoitlxbkr9cysppnld8r+lbvtvmecbhimtuay+uarvcatqoaewkoabpiin5955dl6c1/lozllkjsafog/fgj6g5w==</latexit> (G1 ) v G1 G2 G2 <latexit 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sha1_base64="m8nluskba0f1gqodlz2a3q63488=">aaacenicbvdlsgmxfm34ah1fvzdugkwomzjtef0woiybfuazlaymuwbmmsgjfmoq/9ct/of7sstp+apigu/whraqvspba7n3mo9ouhcqnko826trw9sbhwk2/bo7t7+qenwqk3ivglswjglztdaijaqsettzug3kqtxgjfomlqe+z0xkyrg4k5peujznba0ohhpi/w8aeicvw76tfn+qexunrxwlbgluq7d5udhsdr6je+vddgksdcy4au6rloov0msu0xi1pbsxvjeb6haekzkhanys/yk6fwzcghjgjpntawv38nmssvmvdathkkh2rzm4n/eb1ur1d+rkwsailwffgumqhjops/dkkkwlojiqhlam6feigkwtq0zht5matjrawdnasmp9o4pfpbloquv7jk2rwq61tdpmmrduyoghnwcirabzegdm5ba7qabjf4ai/gybq3nq063u+umytmsfgd6y3b4pjoog=</latexit> (G2 ) 8.! v 8.!? v 8.?!? v? 18 GSF Statics Evidence-based Dynamics <latexit 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sha1_base64="evnh0f2rsf65sccgr7zztuhoba=">aaacb3icbvdnsgmxgpzw31r/qh69bivgxbirgp6k1ivektgf6c4lm2bb0gyyjlmhlh0an8crpoe38epj+ac+h9m2b20dcawz35bhwoqzbvz3y1lalldwy9sfde3tnd2s3v7ts1trwidsc5vo8sacizowzddattrfmchp61wej37rueqnjpi3owsgss4l1jecdzwcvxbyddpffjruvjuqexw3anqivfmpawz1lulb78nsrptyqjhwnc8nzfbhpvhhnnx0u81ttaz4j7twcpwthwqtuqp0bfveiisyj5byql+tmq41nouh/yyxmag571c/m/rpca6ddimktrqqayfrslhrqj8adrjihldr5zgopjtisgak0ym3anot4jzj0zmshwjpba7epmrljlmwcvzk97deblamy1ugem4ghpw4akqcan1aacbb3igf3h1npw35935mj4uobpmafyb8/kdk4+zqa==</latexit> <latexit sha1_base64="6uqcw1azmfmc7kwwadhkcnfhla8=">aaacehicbvdnssnagnz4w+tf/ll5wsycf0migp6k6evvfwwbaelzbdbt0s0m7g6egvospoffqjv4tu38af8dzdpdto6sddmfmmoe6smsmb8bc4tlyymptrb6+sbm1be7sdmsscuzaoggjcamkcaoctbvvjlipicgogokgo+vc7z4qiwnc79u4j6mbpxgfcolpb6570wjqixb14iuplme3i1fbnhw3yjoe+cijrahvbf/pbcbgcx4qozjgpsvpl50goihmz1l1mkhtherqqnqycxut6edl+ao+0ekjdqz+uykn+tuqolnicb/oyrmooz71c/m/rzsq68hpk00wrjqcfrrmdkohffdckgmdfxpoglkjucveqcysvhqzulce8jbhltbhltvquzuwk86rzajm25dydnzp1ui1caaowtfwwdloghvqam2awsn4bi/g1gy3ox342n6umbumt3wb8bndxtbnde=</latexit> 8.! Int <latexit sha1_base64="njlw7iszzxblwekvo9mpwdak0xa=">aaacdnicbvdnssnagnz4w+tfrecvi0wykhe0jmwvisyntae8pms2mbjzhdyow0hfwcbzqe3gtr76cd+b7uelz0nabhwhmg3ayigvuktv+mpawv1b1msb9c2t7z1dc6/rlukmmonghccdzakjhlsuvqx4qacodhgpbembwq/90cepam/v5ou+deachprjjswbmbdixkbgioubv14cgm9q4hztc27bfwktkwaoej7yh57yykzmhcfgzky79ip8nmkfmwmtotejkmk8bgnsv9tjmii/bzspovhwgmhlqefv7bufydyfes5iqn9gsm1kvneif7n9tmvfg55wmmcmezj6kmqzayggyukgwyhnnebzud4v4hatcss9v98pghpkiclpmjad6f2d+huspbuc23luzpqt62qhgjgah+ayooactmataimowoarpim8go8gw/gu/exo10yqsw++apj8wdoe5sx</latexit> 8.?! 17 Int! Int?!? Int! Bool 8.! T1 = T2 <latexit sha1_base64="28/v271yh1had18caf27k5uwamc=">aaaca3icbvdlssnafj3uv62vqks3g0vwvzii6eyounfzos9pq5hmb9qhk0mymqgldokunuvccdu/ra/wp9wkmahrqcgdufcw71z/jgzpw37yyqtrw9sbpw3kzu7e/sh1cojroossafdix7jvk8uccago5nm0i8lkndn0pont5nfewspwctaehadg5kxyagjrbvpoe05+bq3vyzrdl1owdeju5baqhay6t+d0crtuiqmnki1mcxy+2mrgpgocwrw0rbtoiujgfgqcahkdfnd57jm6omcbbj84tgufo7kzjqqvnom8mq6ila9jlxp2+q6odktzmiew2clhyfccc6wtnv8yhjojrpdcfumnmrphmicdwmo8owd6yjcjhgwt9qbnpxlltyjd1g3bhrzv1frltnfrgj+gunsmhaimukmt1eeuhegzvab68l6s96tj8voysoyx+gprm8fonsra==</latexit> <latexit sha1_base64="r4uapsv0nm/nyj8skic74vbahtc=">aaachhicbvdnssnagnzuvxr/oj16wsycf0viru9a8okxgv2bjptnztmu3wzc7qyqql9fr/oe3ssr4gv4bg7scno6sddmfmmo4yemsmbn0ztb1jc6u+be7s7u0fwidhprmnapmujlksbj6shffouooqrgajicjygen709vc78+ikdtmdyplibehmachxuhpawq13dawide4aeh3bp5fwwecwu27zzeaq8spsbnu6iyslzeicrorrjbdug4do1fejosimjg56aasjahp0zgmneuoitlly/jzekqvaoow+nefs/v3ikerlfnk68siqylc9grxp2+yqvdkyylpuku4nwupgyqgbzlwiakghlnefyun0v4gksccu9l+mwwtwgiew02erof7rzqoemhslznmsv9c5ajt1y7u1mu13nvqfh4ascaqdcgja4ax3qbrhk4ak8gxfj0g13oz3xwnnqdin8afgxzdddkb+</latexit> sha1_base64="kdqwzaxuuf2fwfat9u4dwj98vwu=">aaachhicbvblswmxgpy2aq31tdqjl2arvfh2vehjc148vrap6jaszwbb0gx2sbkfsht8iaj/r/exkvg3xc8mz4ebr0iddpfkgh8hdolhefdyq2srucxvfza3tnv17b7+h4lqswicxj2lx4pyjmhdm81pk5eurz6nt9wnfgbqyovi8wthiw0e+geycejwbupa5e8mjayc9sqio8snvygfurazafitigwitsn5wr+/usoagpd+9mlypjgvgjcsvjt10l0j8nsm8lpuoiliiaydhcptg0vokkqk03lj9gruqjkwpgnnjqqvxmzjpqarb65jlduq0vviv7ntvmdnncyjpjuu0fmh4uprzpgkyvqwcqlmo8mwuqy0xwrppayaln0zsgs4cgtldjvmr8ow0dni6akdzfszzj47tioh3xqxvhrkkcachcawuneevrqegdsawgkd4gmfrwqxq232wnommdk8afw+ze2qklv</latexit> sha1_base64="rmahb51bel/zkvrpbzffgu+kkvy=">aaachhicbvbps8mwhe2nm3p+q+7ojtgel47wi5504mjblcv1jhsnn3c0rqk6acuv0uolcfp4y38sr4nqtvzn8e3wqelz3e+txvjhrqszrwyisrbzla+1ysbm1vaoubvlleimgnhiec8zakjhlsulqx4ssconbjpoonlid+z0yepbg/uwlmeieacbpqjjsw+mbvdskbginohbo8pgcorbv1qy6nqvcjvac1bqluy/+ffts9s1p149wehkumensdm0rvr0mcuuxi3nftssjer6haelqylfizc+bls/hovz8qfvoxxwcqr8tgqqltenp4zidewinxh/87qjcs56gevxogjhs4+cheevwcks0kecymvstrawvhefeigewkrvvgnwcwnaev0spmf7sxt/okhslengszte/qtlw3r/vadtbdgeyda3aebhakguakneelyjccb/ainox748v4nd5mpwvjnqmcpzdevwfjv6p6</latexit> <latexit sha1_base64="hvs+aue5spprb5br4cvahwivzq8=">aaacdicbvdnssnagnzuv1r/uj16wsycf0migp6k6mvjbdsgmla2m027dlmjuxulhd6dt+bvn8cbepuzfadfw02ag7yolawz37ddbcmjutn2l1fbwv1b36hvnra2d3b3zoz+tyazwksle5yin0csmmpjv1hfijskgukak4wusn8/gmrkib8k1t4sdoxglemvjagppnl0oeygy6fnth6rv0h2bltuwscjk4fwmbcp2h+e2fcc5iwhvmsmqby6fkz5fqfdmya3izjcncezqia005ion087l6db5rjys6g35cwvl9nchrlou0dvrljnrylnqf+j83yfr06eeup5kihm8/ijigvqklhwbibcgkttvbwfddfeixeggrvvbdk4n5sclkabgunoldnmuvlknvzhjsy7k7b7wvq4q4bacgrpggavqbrega7oag0fwdf7aq/fkvbnvxsf8tgzumqpwb8bnd7mwmuq=</latexit> v static semantics follow via Galois connection (no tweaking!) v 8.!? 8.?! Int <latexit sha1_base64="kqkd7b+lv7phevotpge2v2qjeik=">aaaceicbvdnssnagnz4w+tfvdx5wsycf0sigp606evvfewpnkfstl/apztn2n0ijfqpfakv+gtexktp4ap4hm7thlr1yggy+yydjkg4u9pxvqyfxalldswnl9y3nr297zbao4lrqanoaxbadeawccgpppdu1eaokcdq1gedpxw48gfyvfgx4l4eekl1jiknfg6tr7hhlwjluv7f3hu8uscndneuofunb54nbkeqqec9a397vzimeqhnovgq4zqj9jminamcxmuvvzaqoir96bgqsatkz/l6y3xklb42pcwtgufq70rgiqvguwaui6ihatabip95nvshf37grjjqeht6uzhyrgm82ql3masq+cgqqiuztedeemonouvvtyy9sbkgk3wumozizu7wjxpnlzdp+ren1vq18vcjsadtexcte5qqfbvecnrfggntelerwerdfr3fqyni5yrwyp/yh1+qoxgjx+</latexit> System FG GSF keep track of witnesses of consistent judgments consistent transitivity: combine evidence or fail evidence-augmented term justifies Bool ~? justifies? ~ Int consistent transitivity conservative extension of System F explicit polymorphism natural precision & consistency 19 20

6 GSF Dynamics GSF Dynamics Refinements for GSF type names in evidence: path of bindings let f :? = Λ.λx:.x System F G GSF strengthen CT to enforce parametricity f [Int] hint, Int i <latexit sha1_base64="yf0p8qmgjo0izeqre2xnefi/6dw=">aaace3icbvbltsmwfhtkr5rfgaulnhyvegtujqgjlhvsyfck+pgaudmo01p1hmt2kkoox+aeboee7bbbdsabuado2gw0jgrpppngz55amkq043xzlalldw16npty3nre8fe3euojjwythhcetklkckmctlwvdpse5kgogckg4yvc7/7sksicb/e0h8ga05jshg2kgd+8atmvnuut71ebmj9fbe8nxg152guwiuendg6mcg1sd+9siepzhhgjokvn91hpyzjdfjoq1l1veidxgq9i3lkoykd8rp5ddy6oemeqkovzduv2dyfcs1cqozgsm9ejne44n9dpdtpz5slvboop4uilegdwkingfjjsgytqxcw1lwv4hgscgvtwc0rg1liispp0zcqenhnw1gknbog6ztcu/n682rwubucginwalxwazrgbrrag2cqg2fwal6tj+vnerc+pqmva5bzb39gff4abjgfsw==</latexit> <latexit sha1_base64="yf0p8qmgjo0izeqre2xnefi/6dw=">aaace3icbvbltsmwfhtkr5rfgaulnhyvegtujqgjlhvsyfck+pgaudmo01p1hmt2kkoox+aeboee7bbbdsabuado2gw0jgrpppngz55amkq043xzlalldw16npty3nre8fe3euojjwythhcetklkckmctlwvdpse5kgogckg4yvc7/7sksicb/e0h8ga05jshg2kgd+8atmvnuut71ebmj9fbe8nxg152guwiuendg6mcg1sd+9siepzhhgjokvn91hpyzjdfjoq1l1veidxgq9i3lkoykd8rp5ddy6oemeqkovzduv2dyfcs1cqozgsm9ejne44n9dpdtpz5slvboop4uilegdwkingfjjsgytqxcw1lwv4hgscgvtwc0rg1liispp0zcqenhnw1gknbog6ztcu/n682rwubucginwalxwazrgbrrag2cqg2fwal6tj+vnerc+pqmva5bzb39gff4abjgfsw==</latexit> <latexit sha1_base64="yf0p8qmgjo0izeqre2xnefi/6dw=">aaace3icbvbltsmwfhtkr5rfgaulnhyvegtujqgjlhvsyfck+pgaudmo01p1hmt2kkoox+aeboee7bbbdsabuado2gw0jgrpppngz55amkq043xzlalldw16npty3nre8fe3euojjwythhcetklkckmctlwvdpse5kgogckg4yvc7/7sksicb/e0h8ga05jshg2kgd+8atmvnuut71ebmj9fbe8nxg152guwiuendg6mcg1sd+9siepzhhgjokvn91hpyzjdfjoq1l1veidxgq9i3lkoykd8rp5ddy6oemeqkovzduv2dyfcs1cqozgsm9ejne44n9dpdtpz5slvboop4uilegdwkingfjjsgytqxcw1lwv4hgscgvtwc0rg1liispp0zcqenhnw1gknbog6ztcu/n682rwubucginwalxwazrgbrrag2cqg2fwal6tj+vnerc+pqmva5bzb39gff4abjgfsw==</latexit> <latexit sha1_base64="yf0p8qmgjo0izeqre2xnefi/6dw=">aaace3icbvbltsmwfhtkr5rfgaulnhyvegtujqgjlhvsyfck+pgaudmo01p1hmt2kkoox+aeboee7bbbdsabuado2gw0jgrpppngz55amkq043xzlalldw16npty3nre8fe3euojjwythhcetklkckmctlwvdpse5kgogckg4yvc7/7sksicb/e0h8ga05jshg2kgd+8atmvnuut71ebmj9fbe8nxg152guwiuendg6mcg1sd+9siepzhhgjokvn91hpyzjdfjoq1l1veidxgq9i3lkoykd8rp5ddy6oemeqkovzduv2dyfcs1cqozgsm9ejne44n9dpdtpz5slvboop4uilegdwkingfjjsgytqxcw1lwv4hgscgvtwc0rg1liispp0zcqenhnw1gknbog6ztcu/n682rwubucginwalxwazrgbrrag2cqg2fwal6tj+vnerc+pqmva5bzb39gff4abjgfsw==</latexit> h Int, Inti <latexit sha1_base64="mgjm/8nykwadlczmh11fyn96bne=">aaace3icbvbltsmwfhtkr5rfgaulnhyvegtujqgjlhvsyfck+pgaudmo01p1hmt2kkoox+aeboee7bbbdsabuado2gw0jgrrnpngz55amkq043xzlalldw16npty3nre8fe3euojjwythhcetklkckmctlwvdpse5kgogckg4yvc7/7sksicb/e0h8ga05jshg2kgd+8atmvmqeyp0khm3oen5t2w607dkqeitsjdtbda2b/e2gc05hwjrlsqu86qvszkppirvkalyoieb6jiekbylfmlj+vh8jhsvfcgcshk5hqf5ozchwahihzjjgeqtmvul8z+unorr0m8pfqgnh00vryqboynegdkkkwlojiqhlat4k8qhjhlproavwswkeew06esvvbjzlsyszlnddrru3m9etvrqaoowre4as64ae1wa1qgdtdiwtn4aa/wk/vmvvsf09gkncvsgz+wpn8ac6fsw==</latexit> <latexit sha1_base64="mgjm/8nykwadlczmh11fyn96bne=">aaace3icbvbltsmwfhtkr5rfgaulnhyvegtujqgjlhvsyfck+pgaudmo01p1hmt2kkoox+aeboee7bbbdsabuado2gw0jgrrnpngz55amkq043xzlalldw16npty3nre8fe3euojjwythhcetklkckmctlwvdpse5kgogckg4yvc7/7sksicb/e0h8ga05jshg2kgd+8atmvmqeyp0khm3oen5t2w607dkqeitsjdtbda2b/e2gc05hwjrlsqu86qvszkppirvkalyoieb6jiekbylfmlj+vh8jhsvfcgcshk5hqf5ozchwahihzjjgeqtmvul8z+unorr0m8pfqgnh00vryqboynegdkkkwlojiqhlat4k8qhjhlproavwswkeew06esvvbjzlsyszlnddrru3m9etvrqaoowre4as64ae1wa1qgdtdiwtn4aa/wk/vmvvsf09gkncvsgz+wpn8ac6fsw==</latexit> <latexit sha1_base64="mgjm/8nykwadlczmh11fyn96bne=">aaace3icbvbltsmwfhtkr5rfgaulnhyvegtujqgjlhvsyfck+pgaudmo01p1hmt2kkoox+aeboee7bbbdsabuado2gw0jgrrnpngz55amkq043xzlalldw16npty3nre8fe3euojjwythhcetklkckmctlwvdpse5kgogckg4yvc7/7sksicb/e0h8ga05jshg2kgd+8atmvmqeyp0khm3oen5t2w607dkqeitsjdtbda2b/e2gc05hwjrlsqu86qvszkppirvkalyoieb6jiekbylfmlj+vh8jhsvfcgcshk5hqf5ozchwahihzjjgeqtmvul8z+unorr0m8pfqgnh00vryqboynegdkkkwlojiqhlat4k8qhjhlproavwswkeew06esvvbjzlsyszlnddrru3m9etvrqaoowre4as64ae1wa1qgdtdiwtn4aa/wk/vmvvsf09gkncvsgz+wpn8ac6fsw==</latexit> <latexit sha1_base64="mgjm/8nykwadlczmh11fyn96bne=">aaace3icbvbltsmwfhtkr5rfgaulnhyvegtujqgjlhvsyfck+pgaudmo01p1hmt2kkoox+aeboee7bbbdsabuado2gw0jgrrnpngz55amkq043xzlalldw16npty3nre8fe3euojjwythhcetklkckmctlwvdpse5kgogckg4yvc7/7sksicb/e0h8ga05jshg2kgd+8atmvmqeyp0khm3oen5t2w607dkqeitsjdtbda2b/e2gc05hwjrlsqu86qvszkppirvkalyoieb6jiekbylfmlj+vh8jhsvfcgcshk5hqf5ozchwahihzjjgeqtmvul8z+unorr0m8pfqgnh00vryqboynegdkkkwlojiqhlat4k8qhjhlproavwswkeew06esvvbjzlsyszlnddrru3m9etvrqaoowre4as64ae1wa1qgdtdiwtn4aa/wk/vmvvsf09gkncvsgz+wpn8ac6fsw==</latexit> (f [Int] 1) seal unseal f [Bool] unseal on the outside f [Int] proper handling of type instantiations wrt? GSF Properties Parametricity is incompatible with DGG Type safe, conservative extension of System F f :.! = Λ.λx:.x:: v f :.?! = Λ.λx:?.x:: Well-typed terms are relationally-parametric [Ahmed+17] Dynamic gradual guarantee cannot be satisfied together with parametricity (pick one) must necessarily fail when applied proof independent of representation of evidence and of the interpretation of unknown type 23 24

7 GSF: System F imprecise embedding Gradual Free Claims Disproved imprecise termination has interesting consequences A System F term ascribed to an imprecise GSF type hereditarily terminates.? constant or fail [POPL 11], or unusable result [STOP 09,ICFP 17].? always fail [ICFP 17] new property, weaker than DGG the gradual medium is harmless id ::.? id ::.? are both well-behaved identity functions! Cheap Theorems when looking at the types is not interesting enough (v:.? ) (v = Λ.λx:?.t) always fails Conclusions independent of body 27 28

8 PBC Ahmed et al. POPL 11 Ahmed et al. ICFP 17 FG / FC Igarashi et al. ICFP 17 ie et al. ESOP 18 language cast calculus cast calculus State-of-the-art polym. explicit* explicit* static relations ad-hoc compatibility ad-hoc compatibility ad-hoc language explicit* precision & consistency language implicit consistent subtyping parametricity static GG dynamic GG conjecture - - faithful instantiation s proven (future) (future) conjecture (FC) inherited () proven* conjecture proven - PBC Ahmed et al. POPL 11 Ahmed et al. ICFP 17 FG / FC Igarashi et al. ICFP 17 ie et al. ESOP 18 language cast calculus cast calculus polym. explicit* explicit* Contribution static relations ad-hoc compatibility ad-hoc compatibility ad-hoc language explicit* precision & consistency language implicit consistent subtyping parametricity static GG dynamic GG conjecture - - faithful instantiation s proven (future) (future) conjecture (FC) inherited () new criteria violated by all existing work proven* conjecture proven - GSF language explicit precision & consistency proven proven proven impossible GSF language explicit precision & consistency proven proven proven impossible sensible, systematic design (derived with AGT) can swap Contributions beyond GSF Perspectives 1. (and System FC) are not appropriate internal languages of gradual parametricity 2. parametricity is incompatible with DGG so is noninterference! 3. novel: imprecise embedding of F, disprove claims (language with references) [TOPLAS] Cast calculus for GSF prove equivalence with evidence-based semantics Add implicit polymorphism combine with [ie et al.] Novel form of parametricity to reconcile with DGG? 31 32